The Hardy space $\mathbb{H}_{2}(U)$ on the upper half-plane $U$ is defined to be the space of holomorphic functions $f$ on $U$ with bounded norm,
$$ \|f\|_{H_{2}} = \sup_{y>0} \left ( \int|f(x+ iy)|^2\, \mathrm{d}x \right)^{\frac{1}{2}}$$
The Hardy space $\mathbb{H}^{n\times 1}_{2}(U)$ is the Hardy space of $n\times 1$ matrix valued functions with entries in the space $\mathbb{H}_{2}(U)$.
$(\mathbb{H}^{n\times 1}_{2}(U))^{\bot}$ is its orthogonal complement :=$\{f: f^{\#}\in \mathbb{H}^{1\times n}_{2}(U)\}$ where $f^{\#}(z):=f^{*}(\bar{z})$
The Schur class $S^{n\times n}_{in}$ is the class of inner holomorphic $n\times n$ matrix functions on $U$ and satisfy the condition $I_{n}-s^{*}(z)s(z)\geq 0$.
The problem is the following:
Given $s\in S^{n\times n}_{in}$ with $\det(s)\neq 0$.
Is it true that $f^{\#}(s^{-1})^{\#}\in \mathbb{H}^{1\times n}_{2}(U)$ for every $f$ such that $f^{\#}\in \mathbb{H}^{1\times n}_{2}(U)$?
*Is the problem related to the Smirnov maximum principle?
Thanks in advance!